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A search for the rare decays of $J/\psi \to D_s^-\rho^+ + c.c.$ and $J/\psi\to \overline{D}{}^0\overline{K}{}^{\ast0} + c.c.$ is performed with a datasample of 225.3 million $J/\psi$ events collected with the BESIII detector. Noevident signal is observed. Upper limits on the branching fractions aredetermined to be $\mathcal{B}(J/\psi \to D_s^-\rho^+ + c.c.) < 1.3 \times10^{-5}$ and $\mathcal{B}(J/\psi \to \overline{D}{}^0\overline{K}{}^{\ast0} +c.c.) < 2.5 \times 10^{-6}$ at the $90\%$ confidence level.

Search for the rare decays $J/\psi \to D_s^-\rho^+$ and $J/\psi \to \overline{D}{}^0\overline{K}{}^{\ast0}$