The Exclusive Decay of Upsilon into $h_c$, the $X(3940)$ and $X(4160)$

In this paper, we study double charmonia production in Upsilon peaks, especially, a S-wave charmonium ${\eta }_c$ and a P-wave charmonium $h_c{(}^1{P}_1)$, or a S-wave charmonium $J\mathrm{/}\psi$ and the $X(3940)$ and $X(4160)$ within the nonrelativistic QCD (NRQCD) approach which is a powerful tool to realize the factorization of double charmonia production in electron-positron annihilation. The $J^{PC}=1^{--}$ state $\mathrm{{\rm Y}}(nS)$ can provide an ideal laboratory for studying the properties of double-heavy quarkonium and also can separate the perturbative and nonperturbative parts due to the large heavy quark mass compared with the typical hadron scale ${\mathrm{\Lambda }}_{QCD}$. Explanation of the $X(3940)$ and $X(4160)$ as the $3^1{S}_0$ and $4^1{S}_0$ states, respectively, are compatible with the observed upper limits for the branching fractions of $\mathrm{{\rm Y}}(1S,2S)\to J\mathrm{/}\psi +X$, where $X=X(3940)$, $X(4160)$ by the Belle Collaboration. The branching fractions of $\Upsilon(1S,2S,3S)\to \eta_c+h_c(^1P_1)$arepredictedtobearound$10^{-6}$, which shall be tested in Belle-II experiments.