Inclusive ${\chi }_{cJ}$ ($J=0$, 1, 2) production from $\mathrm{\Upsilon }(1S)$ decay is studied within the framework of nonrelativistic QCD (NRQCD) factorization at leading order in $v_Q^2$, which includes the contributions of $b\overline{b}({{}^{3}S}_1^{[1]})\to c\overline{c}({{}^{3}P}_J^{[1]})+X$ and $b\overline{b}({{}^{3}S}_1^{[1]})\to c\overline{c}({{}^{3}S}_1^{[8]})+X$. 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 $\mathrm{\Upsilon }(1S)$ branching fractions to ${\chi }_{c1}$ and ${\chi }_{c2}$, we obtain the color-octet long-distance matrix element (LDME) $⟨{\mathcal{O}}^{{\chi }_{c0}}({{}^{3}S}_1^{[8]})⟩=(4.04\pm 0.47_{-0.34}^{+0.67})\times {10}^{-3}{\mathrm{GeV}}^3$, where the first error is experimental and the second one due to the renormalization scale dependence, if we use as input $⟨{\mathcal{O}}^{{\chi }_{c0}}({{}^{3}P}_0^{[1]})⟩=0.107{\mathrm{GeV}}^5$ 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 $\mathcal{B}(\mathrm{\Upsilon }\to {\chi }_{cJ}+X)$.

• Received 31 August 2018

DOI:https://doi.org/10.1103/PhysRevD.98.074005