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.
Categories: hep-ph hep-ex