Inclusive $χ_{cJ}$ production in $Υ$ decay at $\mathcal{O}(α_s^5)$ in NRQCD factorization
Inclusive $χ_{cJ}$ $(J=0,1,2)$ production from $Υ(1S)$ decay is studied within the framework of nonrelativistic QCD (NRQCD) factorization at leading order in $v_Q^2$, which includes the contributions of $b\bar{b}({}^3S_1^{[1]})\to c\bar{c}(^3P_J^{[1]})+X$ and $b\bar{b}({}^3S_1^{[1]})\to c\bar{c}(^3S_1^{[8]})+X$. For both channels, the short-distance coefficients are calculated through ${\cal O}(α_s^5)$, which is next-to-leading order for the second one. By fitting to the measured $Υ(1S)$ branching fractions to $χ_{c1}$ and $χ_{c2}$, we obtain the color-octet long-distance matrix element (LDME) $\langle\mathcal{O}^{χ_{c0}}({}^3S_1^{[8]})\rangle =(4.04\pm0.47_{-0.34}^{+0.67})\times10^{-3}$ GeV$^3$, where the first error is experimental and the second one due to the renormalization scale dependence, if we use as input $\langle\mathcal{O}^{χ_{c0}}({}^3P_0^{[1]})\rangle=0.107$ GeV$^5$ as obtained via potential-model analysis. Previous LDME sets, extracted from data of prompt $χ_{cJ}$ hadroproduction, yield theoretical predictions that systematically undershoot or mildly overshoot the experimental values of $\mathcal{B}(Υ\to χ_{cJ}+X)$.