The electromagnetic decays of $X(3823)$ as the $ψ_2(1^{3}D_{2})$ state and its radial excited states
We study the electromagnetic (EM) decays of $X(3823)$ as the $ψ_2(1^{3}D_{2})$ state by using the relativistic Bethe-Salpeter method. Our results are $Γ[X(3823)\rightarrowχ_{_{c0}}γ]=1.2$ keV, $Γ[X(3823)\rightarrowχ_{_{c1}}γ]=265$ keV, $Γ[X(3823)\rightarrowχ_{_{c2}}γ]=57$ keV and $Γ[X(3823)\rightarrowη_{_c}γ]=1.3$ keV. The ratio ${\cal B}[X(3823)\rightarrowχ_{_{c2}}γ]/{\cal B}[X(3823)\rightarrowχ_{_{c1}}γ]=0.22$, agrees with the experimental data. Similarly, the EM decay widths of $ψ_{_2}(n^{3}D_{_2})$, $n=2,3$, are predicted, and we find the dominant decays channels are $ψ_{_2}(n^{3}D_{_2})\rightarrowχ_{_{c1}}(nP)γ$, where $n=1,2,3$. The wave function include different partial waves, which means the relativistic effects are considered. We also study the contributions of different partial waves.