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NRQCD Predictions of D-Wave Quarkonia $^3D_{J}(J=1,2,3)$ Decay into Light Hadrons at Order $α_{s}^{3}$

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 $^3D_J(J=1,2,3)$ quarkonia including D-wave color-singlet, P-wave color-octet and S-wave color-singlet and color-octet are calculated perturbatively at $α_{s}^3$ 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 $ψ(1^3D_J)$ predicted by NRQCD is about $2\sim3$ times larger than the phenomenological potential model results, while for the $b\bar{b}$ system the two theoretical estimations of $Γ(Υ(1^3D_J)\to LH)$ are in coincidence with each other. Our predictions for $ψ(1^3D_J)$ LH decay widths are $Γ(ψ(1^3D_J)\to LH)=(0.43,0.05,0.17)$MeV for J=1,2,3; and for $Υ(1^3D_J)$, $Γ(Υ(1^3D_J)\to LH)=(6.91,0.75,2.75)$KeV for J=1,2,3.