Charmless decays of the spin-2 partner of $X(3872)$

The Belle collaboration recently reported a promising candidate for the spin-2 D^*D̄^* partner of the $X(3872)$, called the X₂ for short, having a mass of (4014.3 ± 4.0 ± 1.5)~MeV and a width of (4 ± 11 ± 6)~MeV. In present work, we assume the X₂ as a pure molecule of the D^*D̄^* under three cases, i.e., pure neutral components (θ = 0), isospin singlet (θ = π/4) and neutral components dominant (θ = π/6), where θ is a phase angle describing the proportion of neutral and charged constituents. Using an effective Lagrangian approach, we calculated the partial widths of X₂→ VV and X₂ → PP (V and P stand for light vector and pseudoscalar mesons, respectively). The predicted decay widths of X₂ → VV can reach a few hundreds of keV, while the decay widths of X₂ → PP are about several tens of keV. In addition, the effects from the proportion of neutral and charged constituent on the decay widths of X₂→ VV and $PP$ are also investigated. We hope that the present calculations will be checked experimentally in the future.