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Lineshape of $e^+ e^-\to D^* \bar D+c.c.$ and electromagnetic form factor of $D^*\to D$ transition in the time-like region

In this work, we apply the vector meson dominance (VMD) model to extract the electromagnetic time-like form factor of the $D^*\to D$ transition combining the recent Belle data for $e^+ e^-\to D^{*+} D^- + c.c.$ and data for $D^*\to Dγ$. Two solutions are obtained in the interpretation of the cross section lineshape: i) With a relatively large coupling for $ψD^*\bar{D}$ determined by experiment, destructive interferences among those charmonium components are required to bring down the overall cross sections, and then account for the cross section lineshape. ii) With a relatively small value for the $ψD^*\bar{D}$ coupling based on heavy quark theory, an apparent cross section deficit near threshold is observed, and contributions from other mechanisms are needed. It might imply the presence of an additional resonance X(3900). Meanwhile, we also point out that an enhancement like that could be produced by the $D_s^*\bar{D_s}+c.c.$ open channel effects.