ηc2 (D2 1) and its electromagnetic decays

The spin-singlet state ηc2(D21) 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[ηc2(1D)→hc(1P)γ]=284 keV, Γ[ηc2(1D)→J/ψγ]=1.04 keV, Γ[ηc2(1D)→ψ(2S)γ]=3.08 eV, and Γ[ηc2(1D)→ψ(3770)γ]=0.143 keV. Considering the strong decay widths are estimated to be Γ(ηc2(1D)→ηcππ)=144 keV and Γ(ηc2(1D)→gg)=46.1 keV, we obtain the total decay width of 475 keV for ηc2(1D), and point out that the full width is very sensitive to the mass Mηc2. In our calculation, the emphasis is put on the relativistic corrections. Our results show that ηc2→hcγ is the nonrelativistic E1 transition dominated E1+M2+E3 decay, and ηc2→ψγ is the M1+E2+M3+E4 decay but the relativistic E2 transition contributes the most.