Pionic transitions of the spin-2 partner of $X(3872)$ to $\chi_{cJ}$

We investigated the pionic transitions between the $X_2$ [spin-2 partner of the $X(3872)$] and $\chi_{c1,2}$ using a nonrelativistic effective field theory. The $X_2$ is assumed to be a bound state of the $D^{*}$ and $\bar{D}^*$ mesons and to decay through several kinds of loops, including the bubble, triangle and box loops. Within the present model, the widths for the single-pion decays $X_2\to\pi^0\chi_{cJ}$ are predicted to be about $3$--$30$ keV. For the dipion decays, the widths are a few keVs. These widths yield a branching fraction of $10^{-3}$--$10^{-2}$. The ratio $R_{\mathrm{c}0}=\Gamma (X_2\to\pi^+\pi^-\chi_{cJ})/\Gamma (X_2\to\pi^0\pi^0\chi_{cJ}) \simeq 1.6$, which is a bit smaller than the expected value of $2$, and $R_{21}=\Gamma (X_2\to\pi\pi\chi_{c2})/\Gamma (X_2\to\pi\pi\chi_{c1}) \simeq 0.85$. These ratios are nearly independent of the $X_2$ mass and the coupling constants, which might be a good quantity for the experiments. Moreover, the invariant mass spectra of the $\pi^0\chi_{cJ}$ final state for the dipion processes are presented, showing a cusp structure at the $D {\bar D}^*$ threshold enhanced and narrowed by the nearby triangle singularity.