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 $\mathrm{\Gamma }[X(3823)\to {\chi }_{{}_{c0}}\gamma ]=1.2$ keV, $\mathrm{\Gamma }[X(3823)\to {\chi }_{{}_{c1}}\gamma ]=265$ keV, $\mathrm{\Gamma }[X(3823)\to {\chi }_{{}_{c2}}\gamma ]=57$ keV and $\mathrm{\Gamma }[X(3823)\to {\eta }_{{}_c}\gamma ]=1.3$ keV. The ratio $\mathcal{B}[X(3823)\to {\chi }_{{}_{c2}}\gamma ]\mathrm{/}\mathcal{B}[X(3823)\to {\chi }_{{}_{c1}}\gamma ]=0.22$, 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 ${\psi }_{{}_2}(n^3{D}_{{}_2})\to {\chi }_{{}_{c1}}(nP)\gamma$, 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.