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We investigate the $J/\psi \phi$ invariant mass distribution of the$e^+e^-\to \gamma J/\psi\phi$ reaction at a center of mass energy of$\sqrt{s}=4.6$~GeV measured by the BESIII collaboration, who concludes that nosignificant signals are observed for $e^+e^- \to \gamma X(4140)$ because of thelow statistics. We show, however, that the three bump structures in the $J/\psi\phi$ invariant mass distribution, though given only by three data points, arecompatible with the existence of the $X(4140)$ state, appearing as a peak, anda strong cusp structure at the $D^*_s\bar{D}^*_s$ threshold, resulting from themolecular nature of the $X(4160)$ state, which also provides a substantialcontribution to the reaction. This is consistent with our previous analysis onthe $B^+\to J/\psi\phi K^+$ decay measured by the LHCb collaboration.  We strongly call for the measurement of this process with more statistics tofurther pin down the nature of the $X(4140)$ and $X(4160)$ resonances.

The $X(4140)$ and $X(4160)$ resonances in the $e^+e^-\to \gamma J/\psi \phi $ reaction