Pionic transitions of the spin-2 partner of $X(3872)$ to χcJ

We investigated the pionic transitions between the X₂ [spin-2 partner of the $X(3872)$] and χc1,2 using a nonrelativistic effective field theory. The X₂ is assumed to be a bound state of the D* and D̄^* 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 X₂→π⁰χcJ 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 Rc0=Γ (X₂→π⁺π⁻χcJ)/Γ (X₂→π⁰π⁰χcJ) 1.6, which is a bit smaller than the expected value of $2$, and R₂₁=Γ (X₂→ππχc2)/Γ (X₂→ππχc1) 0.85. These ratios are nearly independent of the X₂ mass and the coupling constants, which might be a good quantity for the experiments. Moreover, the invariant mass spectra of the π⁰χcJ final state for the dipion processes are presented, showing a cusp structure at the D D^* threshold enhanced and narrowed by the nearby triangle singularity.