An upper bound on $P$-wave charmonium production via the color-octet mechanism

A factorization theorem for $P$-wave quarkonium production, recently derived by Bodwin, Braaten, Yuan and Lepage, is applied to $\Upsilon \to \chi_{cJ} + X$,where$\chi_{cJ}$labelsthe${}^3 P_J$charmoniumstates\mathrm{.}Thewidthsfor$\chi_{cJ}$productionthroughcolor-singlet$P$-waveandcolor-octet$S$-wave$c \bar c$subprocessesarecomputedeachtoleadingorderin$\alpha_s$. Experimental data on $\mathrm{{\rm Y}}\to J\mathrm{/}\psi +X$ is used to obtain an upper bound on a nonperturbative parameter (related to the probability for color-octet $S$-wave $c\bar{c}$ hadronization into $P$-wave charmonium) that enters into the factorization theorem.