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

TL;DR

This paper predicts the mass and electromagnetic decay widths of the yet-undiscovered $ ext{η}_{c2}(^1D_2)$ charmonium state using the Bethe-Salpeter equation, highlighting the importance of relativistic corrections in decay processes.

Contribution

It provides the first detailed theoretical predictions of the properties and decay widths of the $ ext{η}_{c2}(^1D_2)$ state, emphasizing relativistic effects.

Findings

Predicted mass of $ ext{η}_{c2}(^1D_2)$ as 3828.2 MeV.

Calculated total decay width of 475 keV for $ ext{η}_{c2}(^1D_2)$.

Identified dominant decay mechanisms and relativistic corrections in electromagnetic transitions.

Abstract

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