Annihilation Rates of $^3D_2(2^{--})$ and $^3D_3(3^{--})$ Heavy Quarkonia

TL;DR

This paper calculates the annihilation decay rates of specific heavy quarkonium states using the Bethe-Salpeter method, including relativistic corrections, and provides numerical decay widths for charmonium and bottomonium.

Contribution

It introduces a relativistic calculation of annihilation rates for $^3D_2$ and $^3D_3$ heavy quarkonia using the Bethe-Salpeter approach, including wave functions and mass spectra.

Findings

Decay widths for charmonium states: 9.24 keV and 25.0 keV.

Decay widths for bottomonium states: 1.87 keV and 0.815 keV.

Relativistic corrections are significant for high excited states.

Abstract

We calculate the annihilation decay rates of the $^3D_2(2^{--})$ and $^3D_3(3^{--})$ charmonia and bottomonia by using the instantaneous Bethe-Salpeter method. The wave functions of states with quantum numbers $J^{PC}=2^{--}$ and $3^{--}$ are constructed. By solving the corresponding instantaneous Bethe-Salpeter equations, we obtain the mass spectra and wave functions of the quarkonia. The annihilation amplitude is written within Mandelstam formalism and the relativistic corrections are taken into account properly. This is important, especially for high excited states, since their relativistic corrections are large. The results for the $3g$ channel are as follows: $\Gamma_{^3D_2(c\bar c)\rightarrow ggg} = 9.24$ keV, $\Gamma_{^3D_3(c\bar c)\rightarrow ggg}=25.0$ keV, $\Gamma_{^3D_2(b\bar b)\rightarrow ggg}= 1.87$ keV, and $\Gamma_{^3D_3(b\bar b)\rightarrow ggg}= 0.815$ keV.