$η_{c2}(^1D_2)$ and its electromagnetic decays

Abstract

The spin-singlet state $η_{c2}(^1D_2)$ 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 $Γ[η_{c2}(1D)\rightarrow h_{c}(1P)γ]=284$ keV, $Γ[η_{c2}(1D)\rightarrow J/ψγ]=1.04$ keV, $Γ[η_{c2}(1D)\rightarrowψ(2S)γ]=3.08$ eV, and $Γ[η_{c2}(1D)\rightarrowψ(3770)γ]=0.143$ keV. {Considering the strong decay widths are estimated to be $Γ(η_{c2}(1D)\toη_c ππ)=144~\rm{keV}$ and $Γ(η_{c2}(1D)\to gg)= 46.1~\rm{keV}$, we obtain the total decay width of $475$ keV for $η_{c2}(1D)$, and point out that the full width is very sensitive to the mass $M_{η_{c2}}$.} In our calculation, the emphasis is put on the relativistic corrections. Our results show that $η_{c2}\rightarrow h_{c}γ$ is the nonrelativistic $E1$ transition dominated $E1+M2+E3$ decay, and $η_{c2}\rightarrow ψγ$ is the $M1+E2+M3+E4$ decay but the relativistic $E2$ transition contributes the most.