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

Abstract

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 $D^{\ast }$ and ${\bar{D}}^{\ast }$ 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 $D{\bar{D}}^{\ast }$ threshold enhanced and narrowed by the nearby triangle singularity.

Comments: 12 pages, 9 figure, Comments welcome; Correct the last Paragraph of Introduction, accepted by PRD