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 $J/\psi$ 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 $J^{\mathrm{PC}}=1^{--}$ state $\mathrm{\Upsilon }(nS)$ 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 ${\mathrm{\Lambda }}_{\mathrm{QCD}}$. 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 $\mathrm{\Upsilon }(1S,2S)\to J/\psi +X$, where $X=X(3940)$, $X(4160)$ by the Belle Collaboration. The branching fractions of $\mathrm{\Upsilon }(1S,2S,3S)\to {\eta }_c+h_c({}^1{P}_1)$ are predicted to be around $10^{-6}$, which shall be tested in Belle-II experiments.

• Received 17 August 2015

DOI:https://doi.org/10.1103/PhysRevD.92.074017