NRQCD Predictions of D-Wave Quarkonia ${}^3{D}_J(J=1,2,3)$ Decay into Light Hadrons at Order ${\alpha }_s^3$

In this paper, in the framework of NRQCD we study the light hadron (LH) decays of the spin-triplet (S=1) D-wave heavy quarkonia. The short distance coefficients of all Fock states in the ${}^3{D}_J(J=1,2,3)$ quarkonia including D-wave color-singlet, P-wave color-octet and S-wave color-singlet and color-octet are calculated perturbatively at ${\alpha }_s^3$ order. The operator evolution equations of the four-fermion operators are also derived and are used to estimate the numerical values of the long distance matrix elements. We find that for the $c\bar{c}$ system, the LH decay widths of $\psi (1^3{D}_J)$ predicted by NRQCD is about $2\sim 3$ times larger than the phenomenological potential model results, while for the $b\bar{b}$ system the two theoretical estimations of $\mathrm{\Gamma }(\mathrm{{\rm Y}}(1^3{D}_J)\to LH)$ are in coincidence with each other. Our predictions for $\psi (1^3{D}_J)$ LH decay widths are $\Gamma(\psi(1^3D_J)\to LH)=(0.43,0.05,0.17)$MeVforJ=1,2,3;andfor$\Upsilon(1^3D_J)$,$\Gamma(\Upsilon(1^3D_J)\to LH)=(6.91,0.75,2.75)$KeV for J=1,2,3.