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Within the framework of nonrelativisitic QCD factorization formalism, wepresent the next-to-leading-order relativistic corrections to $\Upsilon$exclusive decay into $\eta_c$ plus $J/\psi$. The double charmonia can beproduced 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 obtainedaccurate 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 thetypical heavy quark velocity in bottomonium and/or charmonium rest frame. Thedecay rates are also presented. We find the next-to-leading-order relativisticcorrections to the short distance coefficients as well as the decay rates areboth significant and negative, especially for the corrections from bottomonium.More seriously, the decay rates are even brought into negative by including therelativistic corrections, which indicates the poor convergence for the velocityexpansion in this kind of processes. Detailed analysis is given in the paper.

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