Nonrelativistic QCD predictions ofD-wave quarkoniaDJ3(J=1,2,3)decay into light hadrons at orderαs3

Abstract

In this paper, in the framework of nonrelativistic QCD 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 the $D$-wave color singlet, $P$-wave color octet, and $S$-wave color singlet and color octet are calculated perturbatively at ${\ensuremath{\alpha}}_{s}^{3}$ 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 $c\overline{c}$ system, the LH decay widths of $\ensuremath{\psi}(1^{3}D_{J})$ predicted by nonrelativistic QCD is about $2\ensuremath{\sim}3$ times larger than the phenomenological potential model results, while for the $b\overline{b}$ system the two theoretical estimations of $\ensuremath{\Gamma}(\ensuremath{\Upsilon}(1^{3}D_{J})\ensuremath{\rightarrow}\mathrm{LH})$ are in coincidence with each other. Our predictions for $\ensuremath{\psi}(1^{3}D_{J})$ LH decay widths are $\ensuremath{\Gamma}(\ensuremath{\psi}(1^{3}D_{J})\ensuremath{\rightarrow}\mathrm{LH})=(435,50,172)\text{ }\text{ }\mathrm{keV}$ for $J=1$, 2, 3; and for $\ensuremath{\Upsilon}(1^{3}D_{J})$, $\ensuremath{\Gamma}(\ensuremath{\Upsilon}(1^{3}D_{J})\ensuremath{\rightarrow}\mathrm{LH})=(6.91,0.75,2.75)\text{ }\text{ }\mathrm{keV}$ for $J=1$, 2, 3.