$\hat{I}{\cdot }_{c2}{(}^1{D}_2)$ and its electromagnetic decays

Abstract

The spin-singlet state $\hat{I}{\cdot }_{c2}{(}^1{D}_2)$ has not been discovered in experiment and it is the only missing low-excited $D$-wave charmonium, so in this paper, we like to study its properties. Using the Bethe-Salpeter equation method, we obtain its mass as $3828.2$ MeV and its electromagnetic decay widths as $\hat{I}“[\hat{I}{\cdot }_{c2}(1D)\to h_c(1P){\hat{I}}^3]=284$ keV, $\hat{I}“[\hat{I}{\cdot }_{c2}(1D)\to J/\ddot{I}ˆ{\hat{I}}^3]=1.04$ keV, $\hat{I}“[\hat{I}{\cdot }_{c2}(1D)\to \ddot{I}ˆ(2S){\hat{I}}^3]=3.08$ eV, and $\hat{I}“[\hat{I}{\cdot }_{c2}(1D)\to \ddot{I}ˆ(3770){\hat{I}}^3]=0.143$ keV. {Considering the strong decay widths are estimated to be $\hat{I}“(\hat{I}{\cdot }_{c2}(1D)\to \hat{I}{\cdot }_c\ddot{I}€\ddot{I}€)=144\mathrm{keV}$ and $\hat{I}“(\hat{I}{\cdot }_{c2}(1D)\to gg)=46.1\mathrm{keV}$, we obtain the total decay width of $475$ keV for $\hat{I}{\cdot }_{c2}(1D)$, and point out that the full width is very sensitive to the mass $M_{\hat{I}{\cdot }_{c2}}$.} In our calculation, the emphasis is put on the relativistic corrections. Our results show that $\hat{I}{\cdot }_{c2}\to h_c{\hat{I}}^3$ is the nonrelativistic $E1$ transition dominated $E1+M2+E3$ decay, and $\hat{I}{\cdot }_{c2}\to \ddot{I}ˆ{\hat{I}}^3$ is the $M1+E2+M3+E4$ decay but the relativistic $E2$ transition contributes the most.

22 pages, 4 figures, 6 tables