Predictions of Light Hadronic Decays of Heavy Quarkonium 1D₂ States in NRQCD

The inclusive light hadronic decays of 1D_2 heavy quarkonia are studied within the framework of NRQCD at the leading order in $v$ and up to the order of $\alpha_s^3$. With one-loop QCD corrections, the infrared divergences and Coulomb singularities in the decay amplitudes are proved to be absorbed by the renormalization of the matrix elements of corresponding NRQCD operators, and the infrared finite short-distance coefficients are obtained through the matching calculations. By taking the factorization scale to be 2m_Q, the light hadronic decay widths are estimated to be about 274, 4.7, and 8.8 KeV for the $\eta_{c2}, \eta_{b2}$, and $\eta_{b2}'$ respectively. Based on the above estimates, and using the E1 transition width and dipion transition width for the $\eta_{c2}$ estimated elsewhere, we get the total width of $\eta_{c2}$ to be about 660-810 KeV, and the branching ratio of the E1 transition $\eta_{c2}\to\gamma h_c$ to be about (44-54)%, which will be useful in searching for this missing charmonium state through, e.g., the process $\eta_{c2}\to\gamma h_c$ followed by $h_c\to\gamma\eta_c$.