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In the framework of the nonrelativistic quantum chromodynamics factorizationformalism, we study the processes of $\psi(nS)$ and $\Upsilon(nS)$ decay into alepton pair or a charm pair associated with two jets up to the next-to-leadingorder in velocity expansion. We present the analytic expressions for thedifferential decay rate to the invariant mass of the lepton pair or charm pair.We find that the ratio of the next-to-leading order short-distance coefficientto the leading order one is in the range from -5.5 to -12.4. The relativisticcorrections are so large that they modify the leading order predictionsignificantly. Utilizing the analytic expressions, we also investigate therelativistic corrections in different kinematic regions and their dependence onthe masses of the initial-state quarkonium and the final-state fermion. Inaddition, we study the momentum distribution of $D^{*+}$ in the process$\Upsilon(1S)\to c\bar{c}gg\to D^{*+}X$.

Relativistic corrections to the semi-inclusive decay of $\bm{\psi}$ and $\bm{\Upsilon}$