Relativistic corrections to $\Upsilon$ exclusive decay into double $S-$wave charmonia

Within the framework of nonrelativisitic QCD factorization formalism, we present the next-to-leading-order relativistic corrections to $\Upsilon$ exclusive decay into $\eta_c$ plus $J/\psi$. The double charmonia can be produced through several immediate channels, i.e. $\Upsilon\to g^*g^*g^*\to \eta_c+J/\psi$, $\Upsilon\to g^*g^*\gamma^*\to \eta_c+J/\psi$, and $\Upsilon\to \gamma^*\to \eta_c+J/\psi$. The amplitudes of these three channels are obtained accurate up to ${\cal O}(\alpha_s^3 v^2)$, ${\cal O}(\alpha\alpha_s^2 v^2)$, and ${\cal O}(\alpha\alpha_s^2 v^2)$ respectively, where $v$ indicates the typical heavy quark velocity in bottomonium and/or charmonium rest frame. The decay rates are also presented. We find the next-to-leading-order relativistic corrections to the short distance coefficients as well as the decay rates are both significant and negative, especially for the corrections from bottomonium. More seriously, the decay rates are even brought into negative by including the relativistic corrections, which indicates the poor convergence for the velocity expansion in this kind of processes. Detailed analysis is given in the paper.