Predictions of Light Hadronic Decays of Heavy Quarkonium 1D_2 States in NRQCD

TL;DR

This work computes the inclusive light-hadron decays of the ${}^1D_{2}$ heavy quarkonium family within NRQCD, addressing infrared divergences and Coulomb singularities that arise at ${\rm O}(\alpha_s^3)$ in full QCD. By performing a careful matching between full QCD and NRQCD using the method of regions and operator evolution equations, the authors show that IR-sensitive parts are absorbed into long-distance matrix elements, yielding finite short-distance coefficients. Numerical results for charmonium and bottomonium indicate substantial enhancements from color-octet contributions and next-to-leading order corrections, with ${\Gamma(\eta_{c2}\to LH)\approx 274\ \text{keV}}$ and a total ${\Gamma_{\eta_{c2}}}\approx 660-810\ \text{keV}$, and a large ${\mathcal B}(\eta_{c2}\to\gamma h_c)\approx 44-54\%$, guiding experimental searches through radiative cascades. The findings underscore the importance of color-octet channels in ${}^1D_2$ decays and provide a framework for predicting missing charmonium states in high-energy experiments and near-threshold facilities.

Abstract

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 $α_s^3$. 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 $η_{c2}, η_{b2}$, and $η_{b2}'$ respectively. Based on the above estimates, and using the E1 transition width and dipion transition width for the $η_{c2}$ estimated elsewhere, we get the total width of $η_{c2}$ to be about 660-810 KeV, and the branching ratio of the E1 transition $η_{c2}\toγh_c$ to be about (44-54)%, which will be useful in searching for this missing charmonium state through, e.g., the process $η_{c2}\toγh_c$ followed by $h_c\toγη_c$.

Figures (10)

• Figure 1: Feynman diagrams for ${}^1L_J^{[1,8]}\rightarrow gg$
• Figure 2: Feynman diagrams for ${}^1L_J^{[1,8]}\rightarrow ggg$
• Figure 3: Feynman diagrams for ${}^1L_J^{[1,8]}\rightarrow q\bar{q}g$
• Figure 4: One-loop Feynman diagrams for $(Q\bar{Q})_{^{1}L_{J}^{[1,8]}}\rightarrow gg$
• Figure 5: NRQCD Feynman rules for heavy quark and gluon propagators in different regions