The Excited Charmonium Production in $e^{+}e^{-}$ Annihilation

We calculate the form factor and cross section of the excited charmonium production process $e^{+}+e^{-}\rightarrow \psi \left ( 2S \right )+\eta _{c}$ by light-cone sum rules. In our method, the form factor depends on the distribution amplitude of $\eta _{c}$ meson. Experimentally, the energy scale of $e^{+}+e^{-}\rightarrow \psi \left ( 2S \right )+\eta _{c}$ process is much larger than the initial energy scale of $\eta _{c}$ meson in our BHL model. Therefore, we further consider the evolution of the distribution amplitude with the energy scale, and select the distribution amplitude as our input parameter when the final effective energy scale is $\mu=5.00\ GeV$. This treatment means that we have chosen the relativistic distribution amplitude. The results show that the relativistic effect contributes greatly to the form factor and cross section. Our results are consistent with Belle experimental data.