Annihilation rate of $2^{-+}$ charmonium and bottomonium

TL;DR

This paper calculates the annihilation rates of the yet-unobserved $1^1D_2$ charmonium and bottomonium states using the Bethe-Salpeter method, revealing relativistic effects significantly reduce decay widths and identifying key decay channels for future detection.

Contribution

It provides the first relativistic calculations of double-gamma and double-gluon annihilation widths for $1^1D_2$ charmonia and bottomonia, refining previous non-relativistic estimates.

Findings

Relativistic corrections reduce decay widths by 2-5 times.

Estimated decay width for $1^1D_2 (c\bar c)$ is 432 keV.

Dominant decay channel is $h_c \gamma$ with about 90% branching ratio.

Abstract

The $1^1D_2 (c\bar c)$ state is the ground state of spin-singlet D-wave charmonia. Although it has not been found yet, the experimental data accumulate rapidly. This charmonium attracts more and more attention, especially when the BaBar Collaboration finds that the X(3872) particle has negative parity. In this paper we calculate the double-gamma and double-gluon annihilation processes of $^1D_2$ charmonia and bottomonia by using the instantaneous Bethe-Salpeter method. We find the relativistic corrections make the decay widths of $1^1D_2 (c\bar c)$ 2$\sim$5 times smaller than the non-relativistic results. If this state is below the $D^0{D^{0}}^\ast$ threshold, we can use the sum of annihilation widths and EM transition widths to estimate the total decay width. Our result for $1{^1D_2} (c\bar c)$ with $m=3820$ GeV is $\Gamma= 432$ keV. The dominant decay channel $1^1D_2 (c\bar c)\to h_c \gamma$, whose branching ratio is about 90%, can be used to discover this state.