The lineshape of $\psi(3770)$ and low-lying vector charmonium resonance parameters in $e^+ e^-\to D\bar D$

We investigate the $D\bar{D}$ production in $e^+e^-$ annihilations near threshold in an effective Lagrangian approach. This shows that the lineshape of the cross section near threshold is sensitive to the contributions from $\psi^\prime$, though it is below the $D\bar{D}$ threshold. The recent experimental data from the BES and Belle collaborations allow us to determine the $\psi^\prime D\bar{D}$ coupling constant which appears to be consistent with other theoretical studies. As a consequence, the $\psi^\prime$-$\psi(3770)$ mixing parameter can be extracted around the $\psi(3770)$ mass region. Resonance parameters for $\psi(3770)$, X(3900), $\psi(4040)$, and $\psi(4160)$ are also investigated. The X(3900) appears as an enhancement at around 3.9 GeV in the Belle data. In addition to treating it as a resonance, we also study the mechanism that the enhancement is produced by the $D\bar{D^*}+c.c.$ open channel effects. Our result shows that such a possibility cannot be eliminated.