Inclusive ${\chi }_{cJ}$ production in $\mathrm{{\rm Y}}$ decay at $\mathcal{O}({\alpha }_s^5)$ in NRQCD factorization

Inclusive ${\chi }_{cJ}$ $(J=0,1,2)$ production from $\mathrm{{\rm Y}}(1S)$ decay is studied within the framework of nonrelativistic QCD (NRQCD) factorization at leading order in $v_Q^2$, which includes the contributions of $b\bar{b}({}^3{S}_1^{[1]})\to c\bar{c}{(}^3{P}_J^{[1]})+X$ and $b\bar{b}({}^3{S}_1^{[1]})\to c\bar{c}{(}^3{S}_1^{[8]})+X$. For both channels, the short-distance coefficients are calculated through $\mathcal{O}({\alpha }_s^5)$, which is next-to-leading order for the second one. By fitting to the measured $\mathrm{{\rm Y}}(1S)$ branching fractions to ${\chi }_{c1}$ and ${\chi }_{c2}$, we obtain the color-octet long-distance matrix element (LDME) $\langle$\mathcal{O}$^{\chi_{c0}}({}^3S_1^{[8]})\rangle =(4.04\pm0.47_{-0.34}^{+0.67})\times10^{-3}$GeV$^3$, where the first error is experimental and the second one due to the renormalization scale dependence, if we use as input $⟨{\mathcal{O}}^{{\chi }_{c0}}({}^3{P}_0^{[1]})⟩=0.107$ GeV${}^5$ as obtained via potential-model analysis. Previous LDME sets, extracted from data of prompt ${\chi }_{cJ}$ hadroproduction, yield theoretical predictions that systematically undershoot or mildly overshoot the experimental values of $\mathcal{B}(\mathrm{{\rm Y}}\to {\chi }_{cJ}+X)$.