Hidden charmonium decays of spin-2 partner of $X(3872)$

The Belle collaboration recently reported a promising candidate for the spin-2 $D^*\bar{D}^*$ partner of the $X(3872)$, called the $X_2$ for short, having a mass of $(4014.3 \pm 4.0 \pm 1.5)~\mathrm{MeV}$ and a width of $(4 \pm 11 \pm 6)~\mathrm{MeV}$. Assuming the $X_2$ as a pure molecule of the $D^*\bar{D}^*$, we calculated in detail the hidden charmonium decays of the $X_2 \to J/\psi V$ and $X_2\to\eta_cP$ via the intermediate meson loops, where $V = \rho^0\,,\omega$ and $P= \pi^0\,,\eta\,,\eta'$. The results indicate that the decay widths are strongly dependent on the $X_2$ mass. At present center value of the mass $4014.3~\mathrm{MeV}$, the width for the $X_2\to J/\psi\rho^0$ is predicted to be a few tens of keV, while it is on the order of $10^{2\text{-}3}~\mathrm{keV}$ for the $X_2\to J/ \psi\omega$; the predicted width for the $X_2\to \eta_c \pi^0$ is about a few keV, while the widths for $X_2\to\eta_c\eta$ and $\eta_c\eta'$ are around a few tens and tenths of keV, respectively. We also investigated the dependence of the ratios between these widths on the $X_2$ mass and on the $\eta$-$\eta'$ mixing angle, which may be good quantities for experiments. We hope that the present calculations would be checked experimentally in the future.