The electromagnetic decays of $X(3823)$ as the $\psi_2(1^{3}D_{2})$ state and its radial excited states

We study the electromagnetic (EM) decays of $X(3823)$ as the $\psi_2(1^{3}D_{2})$ state by using the relativistic Bethe-Salpeter method. Our results are $\Gamma[X(3823)\rightarrow\chi_{_{c0}}\gamma]=1.2$ keV, $\Gamma[X(3823)\rightarrow\chi_{_{c1}}\gamma]=265$ keV, $\Gamma[X(3823)\rightarrow\chi_{_{c2}}\gamma]=57$ keV and $\Gamma[X(3823)\rightarrow\eta_{_c}\gamma]=1.3$ keV. The ratio ${\cal B}[X(3823)\rightarrow\chi_{_{c2}}\gamma]/{\cal B}[X(3823)\rightarrow\chi_{_{c1}}\gamma]=0.22$, agrees with the experimental data. Similarly, the EM decay widths of $\psi_{_2}(n^{3}D_{_2})$, $n=2,3$, are predicted, and we find the dominant decays channels are $\psi_{_2}(n^{3}D_{_2})\rightarrow\chi_{_{c1}}(nP)\gamma$, 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.