Inclusive $J/psi$ and $eta_c$ production in $Upsilon$ decay at $mathcal{O}(alpha_s^5)$ in nonrelativistic QCD factorization

We study $J/psi$ and $eta_c$ inclusive production in $Upsilon$ decay within the framework of nonrelativistic-QCD (NRQCD) factorization. In the latter case, for which no experimental data exist so far, we also include the $h_c$ feed-down contribution. We calculate the short distance coefficients completely through $mathcal{O}(alpha_s^5)$. The NRQCD predictions for the branching fraction $mathcal{B}(Upsilonto J/psi+X)$ via direct production, evaluated with different sets of long-distance matrix elements (LDMEs), all agree with the experimental data in a reasonable range of renormalization scale. Using $eta_c$ and $h_c$ LDMEs obtained from $J/psi$ and $chi_c$ ones via heavy-quark spin symmetry, we find that the bulk of $mathcal{B}(Upsilontoeta_c+X)$ via prompt production arises from the $cbar{c}({}^3!S_1^{[8]})$ Fock state. The experimental study of this decay process would, therefore, provide a particularly clean probe of the color octet mechanism of heavy-quarkonium production.