Production of $Y(4260)$ as a hadronic molecule state of $\bar{D}D_1 +c.c.$ in $e^+e^-$ annihilations

We study the $Y(4260)$ production mechanism in $e^+e^-$ annihilations in the framework of hadronic molecules and investigate the consequence of such a picture in different decay channels. In the hadronic molecule picture the $Y(4260)$ is described as a mixture state composed of a long-ranged $\bar{D}D_1(2420)+c.c.$ molecule state and a compact $c\bar{c}$ component. We show that the compositeness relation can still provide a reasonable constraint on the wavefunction renormalization parameter due to the dominance of the molecular component. Such a mechanism can be regarded as a natural consequence of the heavy quark spin symmetry (HQSS) breaking. This study elaborates the molecular picture for the $Y(4260)$ in the $e^+e^-$ annihilations and affirms that the cross section lineshape of $e^+e^-\to \bar{D}D^*\pi+c.c.$ in the vicinity of the $Y(4260)$ should have a nontrivial behavior. In this framework we predict that the upper limit of the $Y(4260)$ leptonic decay width is about 500 eV. We also investigate the coupling for $D_1(2420)\to D^*\pi$ in the $^3P_0$ quark model and examine the possible HQSS breaking effects due to the deviation from the $|^1P_1\rangle$ and $|^3P_1\rangle$ ideal mixing. This will in turn provide a constraint on the HQSS breaking coupling for the $Y(4260)$ to $\bar{D}D_1(2420)+c.c.$ via its $c\bar{c}$ component.