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The $X(4140)$ and $X(4160)$ resonances in the $e^+e^-\to γJ/ψϕ$ reaction

We investigate the $J/ψϕ$ invariant mass distribution of the $e^+e^-\to γJ/ψϕ$ reaction at a center of mass energy of $\sqrt{s}=4.6$~GeV measured by the BESIII collaboration, who concludes that no significant signals are observed for $e^+e^- \to γX(4140)$ because of the low statistics. We show, however, that the three bump structures in the $J/ψϕ$ invariant mass distribution, though given only by three data points, are compatible with the existence of the $X(4140)$ state, appearing as a peak, and a strong cusp structure at the $D^*_s\bar{D}^*_s$ threshold, resulting from the molecular nature of the $X(4160)$ state, which also provides a substantial contribution to the reaction. This is consistent with our previous analysis on the $B^+\to J/ψϕK^+$ decay measured by the LHCb collaboration. We strongly call for the measurement of this process with more statistics to further pin down the nature of the $X(4140)$ and $X(4160)$ resonances.