Su-Yan Pe | Searcharxiv

arXiv2025

Large Relativistic Corrections to Nonrelativistic $M1$ Transitions in Heavy Quarkonium

As double heavy quarkonia, charmonium and bottomonium are generally considered to have small relativistic corrections and can be treated using nonrelativistic models. However, this is not always the case. In this paper, we employ the relativistic Bethe-Salpeter (BS) equation method to calculate the electromagnetic (EM) radiative decays of heavy quarkonium where the $M1$ transition provides the leading-order contribution. Compared to nonrelativistic method which only computes $M1$ transition, our calculations include $M1+E2+M3+E4$ transitions, where the higher-order multipoles, $E2$, $M3$, and $E4$, account for relativistic corrections. The study finds that relativistic effects are large in such transitions even for bottomonium. For instance: the relativistic corrections in the decays $\psi(nS)\rightarrow\gamma\eta_c(mS)$ ($n\geq m$) range from $68.1\%$ to $83.2\%$, while those for $\Upsilon(nS)\rightarrow\gamma\eta_b(mS)$ range between $65.9\%$ and $75.2\%$.

hep-ph↗

arXiv2024

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

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.

hep-ph↗