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

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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 $\alpha_{s}^3$ 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 $\psi(1^3D_J)$ 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 $\Gamma(\Upsilon(1^3D_J)\to LH)$ are in coincidence with each other.