Open charm effects in $e^+e^-\to J/\psi \eta$, $J/\psi \pi^0$ and $\phi\eta_c$

We propose to study the open charm effects in $e^+ e^-\to J/\psi\eta$, $J/\psi\pi^0$ and $\phi\eta_c$. We show that the exclusive cross section lineshapes of these processes would be strongly affected by the open charm effects. Since the final state light meson productions are through soft gluon radiations, we assume a recognition of this soft process via charmed meson loops at hadronic level. A unique feature among these three reactions is that the $D\bar{D^*}+c.c.$ open channel is located in a relatively isolated energy, i.e. $\sim 3.876$ GeV, which is sufficiently far away from the known charmonia $\psi(3770)$ and $\psi(4040)$. Therefore, the cross section lineshapes of these reactions may provide an opportunity for singling out the open charm effects with relatively well-defined charmonium contributions. In particular, we find that reaction $e^+ e^-\to J/\psi\pi^0$ is sensitive to the open charm $D\bar{D^*}+c.c.$ Due to the dominance of the isospin-0 component at the charmonium energy region, we predict an enhanced model-independent cusp effect between the thresholds of $D^0\bar{D}^0+c.c.$ and $D^+ D^{*-}+c.c.$ This study can also help us to understand the X(3900) enhancement recently observed by Belle Collaboration in $e^+ e^-\to D\bar{D}+c.c.$